Here we review the formula of the QCB for multimode Gaussian
states [1]}. In particular, we adapt this formula to our notation
and physical units (here the vacuum noise is \(1/2\) , while in
Ref. [1]} it was equal to 1). Let us consider two Gaussian
states \(\rho \)  (with statistical moments \(X\)  and
\(\sigma \) ) and \(\rho ^{\prime }\)  (with statistical moments
\(X^{\prime }\)  and \(\sigma ^{\prime }\) ). The CMs of
these two states can be decomposed as
\(\sigma  & =S~W(\bar{n}_{1},\cdots ,\bar{n}_{n})~S^{T}\\\sigma ^{\prime } & =S^{\prime }~W(\bar{n}_{1}^{\prime },\cdots ,\bar{n}_{n}^{\prime })~S^{\prime }^{T}, \) 
